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JOURNAL OF ALGEBRA 12, 177-190 (1969)

Semigroup Methods in Ring Theory*

FRANK ECKSTEIN

Mathematisches Institut der Universitrit n/3iinster, 44 Miinster, West Germany

Communicated by A. W. Goldie Received July, 1968

Let A be a ring and denote by R(A) the Jacobson radical. We will write R instead of R(A) if it is clear from the context in which ring we operate. The question of lifting idempotents of the ring A/R to the ring A arises in various contexts in ring theory. However, what one would frequently like to know beyond the existence of idempotents in A mapping onto a given set of idempotents in A/R is whether or not a given set of orthogonal (summable) idempotents of A/R can be lifted orthogonally (summably) to d. In general these questions have a negative answer. There arc rings which contain idempotents modulo the radical which cannot be lifted. There are rings containing orthogonal idempotents modulo the radical which cannot be lifted orthogonally, inspite of the fact that each single idempotent can be lifted. For an example see ([13], 3.A). It will be shown, however, that any countable set of orthogonal idempotents of A/R can be lifted orthogonally to A provided the individual idempotents can be lifted. The approach to the indicated problem will be semigroup theoretical. Extensive use will be made of the structure theory of completely simple semigroups ([5], pp. 76). This procedure has several advantages. Firstly, it leads to more general results, even in very simple cases [compare e.g. ([7], Prop. 5, p. 54) with (Corollary 18). K o reference to the actual lifting is needed]. Secondly, the results are obtained in a more conceptual way. Almost all of the often very tricky calculations can be avoided (compare e.g. [8], Lemma 12, p. 166 and (22) or [13], 4.6 and (21)). Unfortunately our method will not be very helpful in the lifting of single idempotents. It turns out that the lifting of an idempotent is tantamount to the existence of a completely simple minimal ideal in a certain semigroup. In general, however, this is an equally inaccessible problem. On the contrary

* This paper is part of the auther’s dissertation which was written at Tulane University under the supervision of Professor K. H. Hofmann. 177 178 ECKSTEIN

for a special class of semigroups the described correspondence will allow us to conclude that a minimal ideal exists. H. Leptin showed in ([9], Th. 19, p. 260) that any linearly compact ring admits a hyperdirect decomposition into two-sided massive ideals. For rings with identity we can get the same result under a somewhat weaker condition than linear compactness. In ([IO], Th. 2, p. 42) A. Malcev proved that in a finite dimensional algebra which is separable modulo its radical any two complements of the radical are conjugate under an inner automorphism. As another application of the method developed in this paper we will shou- in [6] that this theorem can be considerably generalized.

I. LIFTING 0F Irmr~onm~rS MODULO THE I*.~DICAL

Let S bc a semigroup. A non-empty subset I of S is a left, (u$zt, fwosided) ideal if SIC I (IS C I, SI u IS C I). An ideal K or K(S) is said to be the minimal ideal or the kernel of S if K contains no other ideal properly. A semigroup S is simple if it contains no proper ideals. Let E(S) be the set of idempotents in S, then we introduce a partial order in E(S) by defining e K f if and only if ef = fe = e. An idempotent e is primitive if it is minimal w.r.t. the partial order just defined, i.e. if it is the only idempotent in eSe. A simple semigroup which contains a primitive idempotent is called completely simple. Two elements a, h E S are said to be LP-eqz&aZent, a-P%, respectively .%?-equivalent, a%, if they generate the same left, respectively right ideal. The join of the equivalence relations 2 and .B’ is denoted by ‘9, their intersection by Y. For the algebraic theory of semigroups \ve will refer to the book of Clifford and Preston [5]. All results on semi-groups used in the following can be found there.

I 7'IIIWREM. Let A be a ring, B an ideal contained in the vadicul of A and suppose 0 f e .:- N -+ B is an idempotent in A/B. Then

(a) The residue &ass S =- x + B is a semigroup w.r.t. the multiplication in A as well as w.r.t. the ~-multiplication, .r~y=r$-y~~yanndife~s then e2 eiffe-e =e. (b) For each idempotent e E S, eSe = e + eBe is a group under ovdi?lary multiplication. (c) Each idempotent e E S is contained in the minimal ideal of S which is completely simple. SEMIGROUP METHODS IN RING THEORY 179

(d) Zf e is an idempotent and N = eBe, then e + N” is a normal subgroup ofe$N. (e) The nth term F,(e + N) of the lowev central series is contained in e -+- Nn. Zt1particular, if B is a nilpotent ideal in A the group e + N is nilpotent.

Proof. (a) S is a semigroup w.r.t. the ring multiplication is clear. Let s, f E S then s = x -1. b, t = N $- b’ and

s o t = .Y + (x - A?) + b + b’ f bx -t xb -+ bb’ t .I’ + B.

(1~) Let N E eBe, then s is quasi-regular. Let y be the quasi-inverse of x, theny E eBe and (e - y) is the inverse of (e - ,x) w.r.t. e, i.e., (e - x)(e -y) = e 7. (e - y)(e - x). Thus each element of e -I- eBe has a twosided inverse which means e -I-. eBe is a group.

‘1’0 prove (c) we distinguish two cases:

(I) S = x -/- B contains a zero e. Then x + B contains a minimal ideal, namely e. We have to show e is the only idempotent contained in S = e -1 B. As e is a zero we have e(e +- b) = (e -t- b) e == e. Hence eb = be == 0 for all DEB. Letf- e f b be an idempotent, then e + b = f = f2 = e + b* and h2 = b which means b == 0 as B is contained in the radical of A. Thusf = e. (2) 5’ .:.. m + B does not contain a zero. Then we apply [7] Ex. 14, p. 84 and use (b). (d) First we show e + NT/ is a subgroup. For this we observe that e - u is the inverse of e - Y iff u .‘. r -- ~7%:- 0. Hence if e -- Y E e + NT?, then c: =: rw - Y t -V?l and e - z’ E e + N1. To show that e + N” is normal in c ! iV let e -{ r E e + lVi’ and P $- s, e -+- f E e + IV such that (e s)(e + t) =- P -I- s 2 t + st = e, then

(e + s)(e i r)(e $ t) = e + s -i- r + SY i- t t sf + rt + svf = e A r -t sr + rt + srt E e + N”.

(e) The group r,,(e $ N) are defined as follows: I’,(e + N) = e -+ N and r,,,de + N) - (r,,(e + W, e + 1V). We proceed by induction: The assertion is clear for n = 1. Eow suppose that e + r, e + r E &(e + V) C e -/- A’“, and E T s, e + s E e -$ N, where (e -1 r)(e j- “) = e + Y f~ r -t YY 1.. e and (c $ s)(e -I- S) :: e + s + s + ss y c, then (e $ r)(e --t s)(e + i)(e + S) = (e I- T + s + rs)(e $- i -$ S + 73) = e + f.? + I-S + rFi + SF + sfi f YS + )‘p -1, US f VSG E e + N”+‘. This finishes the proof.

2 COROLLARY. Let A be a complete, topological ring with ideal neighborhoods of zero, B an ideal contained in the radical of A. Let M be a summable set of 180 ECKSTEIN idempotents in A/B and suppose there exists at least one idempotent e E A mapping onto z E i3,iB for each F E &?. Then any collection M of idempotents in A-at most one out of each residue class t?E k-is summable.

Proof. For the definition of a summable family in a topological group vve refer to Bourbaki ([I], pp. 59). It suffices to show that given any ideal neighborhood If of zero, then k’ contains all but finitely many e E M. By hypothesis I- contains almost all E t a module B. Hence for almost all e E M, V A (e 1mB) + ,: Again by hypothesis I,- is a twosided ideal in =1. Thus r n (e -.- B) is a twosided ideal in e + B. Hence the minimal ideal of e j- B is contained in I’ for all but finitely many of the semigroups e + B with e EM. By Theorem 1 this finishes the proof.

3 COROLLARY. Let A be a complete, topological ring with left ideal neighborhoods of zero, B an ideal contained in the radical, let &? be a summable set of idempotents in A/B and suppose there is at least one idempotent e E A mapping onto F E A/B f OYeach .?E M. Then any left ideal neighborhood V of zero cuts almost all minimal ideals K(F), P E M, and hence contains at least one minimal ieft ideal of the semigroups F s -I B for almost all F E Ilil. Proof. Again L = 1’ n (e -t B) is a left ideal of F -:- x + B for almost all FE Ff. Let k’ be the kernel of e -I- B, then KL C K n L C I’.

4 PROPOSITION. Let 9 be a &,a and 13 1111ideal contained in the radical of A. Let e be an idempotent in =IjB and suppose the residue class F contains an idempotent e. Then all idempotents of e + B aye obtained in the following fashion: If b, b’ E B then f = e + eb( 1 - e) and g = f I (1 --f) b’f are idempotents and e9f and fYg. Dually one obtains idempotents f' = e + (1 - e)be and g’ -:f’-I-f’b’(l -f’)ande9f’andf’&‘g’.I n p ar t icu 1ar zf e is contained in the centralizer of B in A, then e $ B contains at most one idempotent.

Proof. By Theorem 1 each idempotcnt in S = e $- B is contained in the minimal idcal K(S) and K(S) is completely simple. By ([5], 2. 5 I, p. 79) K is a V-class and by ([5], 2.49, p. 78) k J is the union of its minimal left, respectively right ideals each of which is an Ye-, respectively an g-class ([.5], 2.43, p. 76). If e, f are two idempotents in the same T-class then ef == and fe = f ([5], 2.14, p. 58). Using the Peirce decomposition of B

B = eBe + eB( 1 - e) + (1 - e) Be + (1 - e) B(1 - e) and the above formula we get f 9e iff

fce+(l -e)Be i.e., f = e -j- (1 - e) be for some b E B. SEMIGROUP AIETIIODS IN RING THIIORY 181

Each element of this form is automatically an idempotent. Similarly w-e get gAj iff g =fi-fb’(l -f) for some b’ E B. U-e already observed that K is a 8-class. The result follows from the structure of K.

5 COROLLARY. The minimal ideal K of e + B is a group iff c is contained in the centralizer of B in d.

Proof. Suppose K is a group. Then j m= e + be - ebe = r. Hence be = ebe for all b E B. Dually we get eb = ebe for all b t B, and thus the two-sided Peirce decomposition degenerates to B = eBe + (1 -. e) B( 1 -- e). This, however shows that e is in the centralizer of B in ,4. The converse is clear from Proposition 4 and the fact that K(e + B) is a union of groups.

6 COROLLARY. Let A be a complete, topological ring with left ideal neighbor- hoods of zero. Then any set of central idempotents is summable in iz iff it is summable modulo the radical.

Proof. This follows immediately from Corollary 3 and 5 and ([I], Prop. 5, p. 65).

7 PROPOSITION. Let the l~ypothcses be as in Proposition 4. If e, g are idempotents ire e + B then there exists arc idempotent f E (e + B) such that g((rf) = g and an idempotent fi such that ,fieg := g. Proof. (egg-box structure). As the minimal ideal is completely simple and a 9-ciass there are always idempotentsj, .fi t (e + B) such that fB’e and fZ?g and fi3?e and ji%g. These idempotents will have the stated properties.

So far we always assumed that single idempotents of A/R could be lifted to A. Before we go on let us pause for a moment to be on the lookout for some classes of rings which satisfy this hypothesis. First of all there is the class of SBI rings which have this property. This notation is due to Kaplan&y. For the definition see e.g. ([7], p. 53). A s we shall show the class of linearly compact rings enjoys this property. For commutative rings which are linearly compact in the discrete topology this was proved in ([12], p. 86).

8 DEFINITION. A topological =I-module M is linearly compact in case every filter of closed A-submodules has adherence points and M has a neighborhood basis of zero of A-submodules. A topological ring A is linearly compact in case it is a linearly compact left A-module. The following lemma about linearly compact modules will be needed. 182 ECKSTEIN

9 LEMMA. (a) A linearly compact A-module is complete. (b) If f is a corztinuous module homomorphism of a linearly compact module M into a topological module M’ with module neig-hborhoods of zero. Then f(M) is linearly compact in M’ and hence closed. Proof. See e.g. ([9], p. 244).

10 PROPOSITION. Let A be a topologicul ring and B a linearly compact ideal in A. If t?is an idempotent in A/B, then there exists an idempotent e in A mapping onto F module B.

Proof. Let !JJJJ1be the set of all multiplicative subsemigroups off + B of the form R $- ilf where f is any element mapping on F modulo B and 34 is a closed submodule of d contained in B. The set ‘351is partially ordered by inclusion 3 and is inductive with respect to this order; for let % be a chain in ‘332then fi 92 f ii since B is linearly compact by assumption. The semi- group f +- B is closed by hypothesis, thus in ‘351and !jJI # g. By Zorn’s lemma w-e pick a minimal element in 5lJ331and denote it by S : g + M. 11-e consider the semi-group 3~ (,y ! AI) s ,$ f Mg. Clearly Sg C S and by Lemma 10 nlg is a closed submodulc of B. Thus g2 I Mg t 911 and g’ $ !l~~~C S. RIoreover, S \vas chosen to be minimal, so that SF S. Therefore thcrc exists an P t A’ such that q .g and (e% -~p) g ~~~0. C’learl\ 3 - e t :lZ. Let ,V be the closed submodule of

Proqf. Suppose f ~ :If is a semigroup, then .f2 .f i- m and f” - ,f~ ilf also (f + ~)f = .f” :~ fflf = f $~ E. Hence n$ ~~ -j” f -$- li t M. The converse is even more trivial. We use this characterization to show that P ~.~A’ is a semigroup. (eZ -~- e) g -:~~0 by the definition of e. Thus e2 ~~ e c X. Also (IE) ,g m_ne ~ ===0 bvI the definition of e and 3. Thus iVe C n: and P -+ !I: is a semigroup which is contained in g -I- 111 S by definition. Using again the minimalit!- of S yields e + N g + .&I = g2 -1 Ilfg -: q -+ l\‘g f;y. Hence f’ g and e2 m.e.

11 COROLIARY. Let 9 be a topological ring and suppose 13 is a compact ideal in A. Then all idempotents of .4/B call be lifted to A4.

Proof. The only place in the previous proof where we used that il has left ideal neighborhoods was when we concluded that A@ is closed whenever M is closed. But this is trivial if R is compact.

12 COROLLARY. Let A be a topological ring with left ideal neighborhoods SEMIGROUP METHODS IN RING THEORY 183 of zero and B an ideal which satisfies the descending chain condition on closed A-left ideals. Then all idempotents of A/B can be lifted to A.

Proof. By ([12], Prop. 5, p. 81) B is linearly compact.

2. SEMIGROUPS WHICH ARE TRANSLATES OF A RING

In (Chapter 1 we associated with a given ring A certain semigroups. We used this correspondence to gain information about the ring A. Now we wish to employ this correspondence in the other direction; namely to gain insight into the structure of a special class of semigroups.

13 DEFINITION. A semigroup S will be called a translate of a ying ;1 if there exists a ring B and an epimorphism f : B --f C onto a simple ring C with identity such that S = f-'(I) and A = f-‘(O). Alternatively, a semigroup S is a translate of a ring A if there exists a ring B containing rZ as an ideal such that S = .X + A~ for some representative x t B of the identity of the simple ring H/.4. An ideal I of a semigroup S is called K-potent if some power I” C K, where K is the minimal ideal of S. The union of all K-potent ideais of S is called the radical R(S).

The concept of a semigroup which is a translate of a ring is a generalization of what E. Clark called a translafe of an algebra ([4], 1.5, p. 435).

14 PROPOSITION. Let S be a translate qf a ring A and B, C as in (13) suclr that B/R(B) z C @ il and idempotents of B/R(B) can be lifted to B. Then

(a) S contains a completely simple minimal ideal K. (b) If the keynel of S is a group, then S is isomorphic to D x G, where I) is a yin‘g and G is a translate vf a radical ring zchiclr is isomorphic to the minimal ideal I;, and conaersely. (c) If S is commutative, then S is isomorphic to D x G, where D is a King and G is a translate of a radical ritlg which is isomorphic to the minimal ideal A-.

Proof. Let A, B be as in Definition 13. The radical R(B) of B is contained in A, since B/A g C is semi-simple. Consequently the epimorphism f associated with S factors through the radical R(B) of B, i.e. f = lz . g.

B/R(B) z C @ A 184 ECI

Under the given hypothesis B/X(B) contains at least one ideal with identity F which is isomorphic to C. TT’e fix one such for the rest of the proof, and denote it again by (J‘. Since C is simple g(A-I) n C’ (0: md B/R(B) g g(A) @ c. Let s be :my representative in B of the identity e of C, then

s .i’ -c- 4 -+ R(H) s : A.

son .I R(B) is a semigroup idcal in S , since s is orthogonal to any n E A module R(B). liy hypothesis any idempotent of B/R(B) can be lifted to B, a-hich means x ! R(B) contains a completely simple minimal ideal K by Theorem 1. But K is also an ideal in S, hence is the minimal ideal of S. Let e be any idempotent in R, then s + R(H) e - R(A), since

R(A) -~- A n R(B) ~~ R(B).

(b) M’e recall from Corollary 5 that A is a group iff its unit element E commutes elementwise with K(A). \Yc also recall from the first part of the proof that eA, Ae C R(A). With these two observations the Peirce decompo- sition of A collapses to

and e .s z-I =:= (e -i eile) -1 (1 -- e) L3(1 e) IS the desired decomposition. To see that eAe is a radical ring we use that R(eAe) 7 eR(-4) e and e.4r C R(A). (c) This is a special case of (b).

I5 PROPOSITIOS. Let S be u translate of a ring A and B, C as in (13) such that B/R(B) -” C 0 A4 and the radical R(B) is nilpotent. Then R(S) = c A- R(A), where e is any idempotent in I<. The radical R(S) is the unique maxinto/ K-jotent ideal of S and (R(S))7’ C K for any n greater than OY equal to the mlpotemy index of R(‘4).

Proof. In order to show that e - R(A) is the radical of S \ve have to show that any K-potent ideal 1 (i.e. an ideal satisfying 1” C K for some n) is contained in c -c R(A). It suffices to shovi

is contained in e f R(A), since I C M(1). Clearly, M(I) is again an ideal in S, since S is a cosct and the multiplication on S is induced by a distributive ring multiplication. Hence u-e may assume I 2 n/r(l). Let e be any idempotent in K, then e c I and we first show that (I - P) is a ring ideal of A. Now A(1 - e) = (,S ~~ e)(I - e) C SI - el -~~Se -1~e C(I-I-l)+E=e-(I+l-I) C.c-I:zI-e. SEMIGROUP METHODS IN RING THEORY 185

Hence A(1 - e) C (I- e) and I - e is a left ideal of A. Similarly, one shows I - e is a right ideal of A. By hypothesis In C e + R(A), hence g(l)n = g(e), where the map g is defined as in (14). g(1 - e) = g(1) - g(e) is an ideal in g(A) C B/R(A) and (g(1) -g(e))” = g(l)” - g(e) = 0, since&e) is orthogonal to every element in g(A). But g(A) is semi-simple and thus contains no nilpotent ideals different from zero. This means g(1 - e) := (01 or I - e C R(A), which means I C e f R(A). To see that e + R(A) is the radical R(S) of S it suffices by n-hat we have already proved to show that e + R(i3) is a K-potent ideal. To do this we have to exhibit an integer n such that any zr-fold product of clcments in e + R(A) is contained in K. For that purpose we prove the following statement. If (e + rr),..., (e f vn) E e + R(A), then

fi (e + YJ E K -;- R(A)l’ i-1

The assertion is clear for n = 1, since e E K. Now we assume the assertion holds true for 11 = K - 1. Then

fi (e + yi) = ( ‘fj (e + ri)) (e --I.-yic) = (s + y)(e -i r,),

where s E K and Y E (R(A))‘,-‘. Such elements exist by induction hypothesis. Since s is contained in the minimal ideal K, it is contained in a minimal left ideal f + R(A)fof e + R(d). Thus there exists a t E R(il)f such that s = f $ t and tf = t. Since e $ R(A) is a residue class and f E e + &?(/I) there exists n r, E R(A) such that e + yl, =f + F~. Now we are reads to compute the product (s A- Y)(E 1 Y,:): In fact,

(s + y)@ + yk) = (f + (t + y))(f -i- Fk)

The element in the first parenthesis is contained in K, since for any idem- potentfg K, we have (f + R(A))f(f + R(A)) = K. It remains to investigate the last term:

since tf --- t. By induction hypothesis Y E (R(A))“-l, hence ran - rffk E R(A)“. With these remarks we have proved, that lJf=, (e + ri) E K -I- (R(A))‘;. This finishes the induction. Again by hypothesis R(A) is nilpotent, hence there exists an integer m such 186 ECKSTEIN that R(A)“’ == (0). If we take n > m, then our formula yields (e + R(A))” C K, hence e + R(i3) is K-potent. This finishes the proof.

E. C‘lark showed in his dissertation ([2], th. 3.5, p. 10) that any affine semigroup is a translate of an algebra. For the definition of an affine semigroup we refer to [3]. Using the \Vcdderburn-Artin theorem about the structure of finite dimensional semi-simple algebras and, c.g. (12), one easily sees that the hypotheses of the propositions (14) and (I 5) arc satisfied in his case. With this remark his results ([3], 1.4, I .lO, 1.11, 2.7) and ([4], 2.4, 2.5, 2.6) are easy consequences of the Propositions (14) and (15). Our method of proof has not only the \irtuc that it is much shorter, but also makes clcarcr what is needed in order to obtain the given results.

3. LIFTING OF ORTHOGONAL IDEMPOTEWS

16 LEMN.\. Let S be a sem@oup with u”eyo which contains two completely simple subsemigroups A, D such that 0 E AD, DA. Then there exist idempotents L-E ,4 andf E D such that ef =-- .fe 0.

l’roqf. By hypothesis there exist a, a’ E ,-1 and d, d’ E 1) such that 0 mm_ad : d’a’. Hence L(a) R(d) = L(d’) &a’) my-{0), where the Y- and ti-classes L(a), K(a’), respectively R(d), L(d’) arc taken in -4, respectively D. Let e be the idempotent in L(a) n R(a’) and f be the idempotent in R(d) n L(d’).

17 PROPOSITION. Let 4 be a ring, B an ideal contained in the radical. Suppose JjB contains afinite number of orthogonal idempotents pi (i = I,..., n). Then the following statements are equivalent:

(1) The idempotents 2, raw be lifted to =1. (2) The idempotents ei can be lifted orthogonally.

Proof. (2) implies (1) is clear. (1) implies (2): We proceed by induction. Suppose we already constructed e, ,..., e,, such that eie3 = Zfje, and ei E ei, i = I,..., m. Then we define e ~~~,r,andu -f-ef-fe f efe, where f is any idempotent in & +l . Then u E f + B -~: e,,,,, and ue eu ===0, hence (fu) e = e(uf) = 0 and fu, uf are contained in the minimal ideal K off + B. Now we are ready to apply the lemma with .4 := {e> and II = K(f + B). Remark. The argument just given shows also that any countable set of orthogonal idempotents of A/B can be lifted orthogonally to A, whenever the idempotents can be lifted individually. SEMIGROUP 2IETHODS IN RING THEORY 187

I8 COROLLARY. Let d be a topological ring, B a closed ideal contained in the radical. Suppose A/B contains a sumnrable set of orthogonal idenzpotents 1~;). Then (I) implies (2).

(1) Theue is an idempotent cross section rztnnin~ through the residue classes 6, and anjj such cross section is summable. (2) The idempotents ei cat1 be lifted ovthoxonally. Proof. ‘The same proof vvorks as above.

19 COROLLARY. Let ,A be a complete, topological ring with ideal neighborhoods of ZPYOand B a closed ideal contained in the radical. Su;hpose -4,/B contains a sunmable family of orthogo?lal idempotents (F ,). Then the statements (1) and (2) are eqz&aiellt.

(1) The idempotents P, can be lifted to -4. (2) The idempotents e, can be lifted orthogonally to L3.

P~wo~. ‘This follows immediatelv from Corollary 2 and Corollarv IX.

20 COROLLARY. I?L a linearly compact I-ing A zoith ideal neighborhoods of zero. Any summable set qf orthogonal idempotents module a closed ideal B, B C R(A), can be lifted orthogottally to -3.

Proof. This follows from Proposition IO and Corollary 19.

21 I’ROPOSITIOh-. Let .-1 be a topological ying, B a compact ideal contained in the radical of 12. Suppose {ei : i E Jj zs‘. a set of orthogonal idempotents in .4/B. Then the idempotents (e,) cau be lifted orthogonally to A.

Before 11-estart the proof we state the nest proposition and then prove them simultaneously.

22 ho~oSlTlOs. Let -3 be a linearly compact rig, B a closed ideal contained iu the radical of ,;I which has the property that er;evy jilter of closed resets of .-l-vl;p-llt-sub3izodules has adherence points. Suppose (ei : i E J) is a fami<~~ of orthogonal idempotents in .4/B. Then there exist orthogonal idempotents e, 5 A mappiq: onto P, .

Proof. Hy Zorn’s lemma choose a maxima1 collection of orthogonal idempotents (e, E .4 : i E I: mapping onto F?modulo B. Suppose I + J. Then choose fj,,. such that 12$ I and let ,f be any idempotent in rl mapping onto F,, module B. \\:e consider the following equations.

(f ~~ S)PI 0 equivalcntlj xef =- fej , .x E B e!(.f - x) 0 equiralentl~ e,.x = e,f, XEB. LetrRi denote the closed coset of solutions in B of e,.y ~= c,f and denote h! Li the closed coset of solutions in B of xr, fe, . Hy Proposition 17 the collection (Ri , L, : ;,j E 1$ has the finite intersection propertv. Thus by our hypothesis R = (),EI R, -$ anJ 1, f),,,l,, f : . Let .x c R and JJ c I,. Then e(f - x) = 0 and (.f ~1)c, 0 for all i E 1. Thus

(f - x).f(f -d ei = 0, Pl(f - 4f(f - ?I) 0 and (f - x)f(f -v) is contained in the minimal ideal of f -: B b! Theorem I. Let e,. he the identity of the group containing (f ~ ,x)f(f ~ J), then elsei := e7el; =: 0 for all it I. Rut this is a contradiction to the maximality if I. Hence I = J. With Proposition I7 in mind one can state ([7], ‘I%. I, p. 55) in a somewhat more general form.

23 THEOREM. Let A be a ring with identity. Suppose A/R(A) E B, is u matrix ring over a ring B and assume idempotents of A/R(A) can be hffed to il. Then ,g is isomorphic to a matrix ring B,, over a ring B, zcjhere B/R(B) pZ B.

Proof. ‘I’hc proof is virtually the same than the one given in ([7], p. 55). One only has to use Proposition 17 instead of ([7], Prop. 5, p. 54).

24 PROPOSITIOK. Let A be a complete, topolo

Proof. Us; Corollary 6 WC may assume that J is semi-simple. As a simple ring contains at most one central idempotcnt the assertion follows from ([I], Prop. 4, p. 65).

25 PROPOSITION. Let ,1 be a complete, topological ring with left ideal neigh- borhoods of zero. Suppose that .g;R(=1) g JJ A, , where the Ai are simple rings with identities. Let E be any set of nonzero, ce)ltval idempotents of .4 satisfyirlg the folloeainx conditions: (a) ~e,fEBande=-.f,thene-ffIE (b) if the orthogonal family {e, F Ej iz. summable, then e :-: C ei E E.

Then to any idempotent e t E there exists an idempotent f ‘g e such that f~ B andf is minimal in this respert. Proof. See ([II], Prop. IO, p. 6). One only has to use (24) instead of his Proposition 7, which is by the way a special case of (2).

26 THEOREM. Let A be a complete. topolqpical ring with identity and with left SEMIGROUP METHODS IN RING THEORY 189 ideal neighborhoods of zero. Suppose fitrthermore A/R(A) 2 n Xi , where z4i are simple rings. Then

(1) A is isomorphic to a direct product JJ d, of indecomposable twosided, closed ideals. (an ideal is cabled indecomposable if in any direct decomposition L-I,. -- B @ C in closed twosided ideals B, Coue of the summands must be zero). (2) .ilny closed onesided ideal I is isomorphic to n (A,( n I) under the nboae isomorphism.

Proof, Let Z be the set of all central idempotents in A. Then Z satisfies the conditions:

(a) ife,JgZande:,f,thene--fzZ, (1~) if an orthogonal family {e, : eL E Zj is summable in ;1, then e -- C ei E Z.

Thus % satisfies the hypotheses of Proposition 25 and hence contains minimal idempotents. By %orn’s lemma we can pick a maximal set 32 of minima1 orthogonal idempotents in Z. Then by Proposition 24 the sum E 2 z {f : f E M} exists. Clearly, e is an idempotent and we claim L ~= e. Suppose 1 + e, then the idempotent I - e E Z, hence there is a minimal idempotent g E Z below 1 - e. Obviously gf = fg = 0 for all f E M, hence 112 cannot be maxima1 which is a contradiction. Thus I = C {f :f~ M}. By the definition of summability 1 =: C {f : f E M} means that the subring B of A generated by the closed ideals (Af : f E n/r> is dense in A. As the idempotents f E IV! are pairwise orthogonal B s C Af, where the sum x ajljis endowed with the topology induced by the product topology. I%ow we apply ([I], Prop. 5, p. 41) to obtain B g n Af. The ideals Af are indecomposable, since the identity of Af is a minimal, central idempotent. (2) Let I be a closed left ideal, say, then If = Af n I, and one concludes just as above that I G IJ {Af n I) : f E -113).

ACKSOWLEDGMENT

The author wishes to express his gratitude to Professor K. H. Hofmann for his \;duable assistance.

1. Boi-RBaKI, N. “Topologie GtnCrale: Lirre III,” Chap. 3. 1960. 2. CI.ARK, E., Affine semigroups over an arbitrary field. Tulane dissertation, New Orleans, Louisiana, 1964. 3. CLARK, E. Affine semigroups over an arbitrary field. Proc. Glasgow Math. dssoc., 7 (1965), 80-92.